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The mechanical properties and relaxation spectrum of aluminum soap-hydrocarbon solutions Flynn, James Thomas 1953

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THE MECHANICAL PROPERTIES AND RELAXATION SPECTRUM OF ALUMINUM SOAP-HYDROCARBON SOLUTIONS by James Thomas Flynn A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of PHYSICS We accept this thesis as conforming to the standard required from candidates for the degree of DOCTOR OF PHILOSOPHY. Members; of the Department of Physics THE UNIVERSITY OF BRITISH COLUMBIA September, 1953 ABSTRACT A sew instrument is described for the measurement of the mechanical properties of polymer solutions and the theory necessary for its application is developed. The instrument has been used to determine the mechanical properties of aluminum soap-benzene gels over a frequency range of six decades* The modulus of rigidity has been found to be of the order of lC^to 1C)4- dynes per square centimeter and the viscosity of of the order of 105 to 10^ poise. Both rigidity and viscosity vary approximately with the cube of the concentration. The peak in the relaxation spectrum is at a relaxation frequency of the order of 3xlC-"3 -/ seconds and has been found to be independent of concentration. It is concluded from the experimental results that the structure is either a three dimensionally cross-linked network of long chain polymer molecules: or a micellar structure with the micelles linked together by common molecular chains. ACKNOWLEDGMENTS The author wishes to express his appreciation to: Dr. G.L. Pickard, who acted as supervisor for this work, especially for his advice and assistance on the preparation of this thesis* Dr. A.M. Crooker, for the loan of optical components and for his generous advice and encouragement. The Defence Research Board for the financing of this research project and especially for generously granting leave of absence and financial assistance to the author for graduate work at the University of British Columbia. Dr. G.M. Shrum, Head of the Physics Department of the University of British Columbia, for permission to carry out this research at the University and for his co-operation i n making i t s successful completion possible. TABLE OF CONTENTS Page INTRODUCTION i CHAPTER I THEORY 1 Phenomenological Theory of Viscoelasticity 1 The Determination of ^ Y#and from Relaxation Measurements 8 The Determination of ^ tf/from Vibration Measurements . . . 11 Theory of the Concentric Cylinder Visco-Elastometer . . . . 14-Theory of the Electro-Mechanical Transducer . . . . . . . . 20 CHAPTER II APPARATUS . o . . . . . . . . . . . . . 24 CHAPTER i n CALIBRATION OF THE VISCO-ELASTOMETER . 30 Measurement of the Mechanical Constants, S, J, and (Blr) 2!©" 9 30 Calculation and Experimental Verification of the C e l l Constant . . . . . . . . . . . . . . . . . 31 CHAPTER IV PROCEDURE . 35 CHAPTER V EXPERIMENTAL RESULTS 33 The Effect of Magnitude of the Strain . 33 Page The Modulus of Rigidity from both Velocity of Transverse Sonic Waves; and Visco-Elastometer Measurements • • • • • » 40 The Dependence of Rigidity on Concentration • 43 The Relaxation Function and Viscosity . . . • 45 The Distribution Function /Vfy) . . 47 CHAPTER VI DISCUSSIOH . . . . . 50 APPENDIX! . . . . . . . . . . . . . . . . . . . 57 BIBLIOGRAPHY: 59 ILLUSTRATIONS To Follow Page 1. a. Maxwell Element b. Parallel Array of Maxwell Elements . . 1 2. Schematic Drawing of the Test C e l l . . . . . 6 3. Mechanical Model of the Visco-elastometer for Relaxation Measurements: . . . . . . . . . . 8 4. The Visco-Elastometer 25 5. The Inner Cylinder . 26 6. The Alternating Current Bridge Circuit 27 7. Graph of_/ *~X* Y e 33 3, Graph of Log. Rigidity vs Log. Concentration for Digel-Benzene Solutions . . . . 44 9. Graph of Angular Deflection of the Inner Cylinder During Relaxation for Digel-Benzene Solutions 45 Plate I The Visco-Elastometer . 58 TABLES Page I Mechanical Constants of the Visco-Elastometer 37 II The Modulus of Rigidity of Octal and Octoic Acid i n Gasoline from Three Experimental Methods • 42 III The Modulus of Rigidity, G, of Digel i n Benzene . . . . . . 44 IV The Relaxation Function for Digel i n Benzene . . . . 46 V The Distribution Function //(>>> for Digel i n Benzene . . . 48 VI The Viscosity Calculated from the Rigidity and Relaxation Frequency of the Predominant Term i n A/W for Digel i n Benzene . . . . . . . . . . . . . . . . . . . 49 INTRODUCTION Solutions of aluminum soaps i n hydrocarbons form relatively stable systems which appear to be two phase, a three dimensional network of solvated material and a dilute soap solution* These solutions have a high viscosity and modulus of r i g i d i t y typical of solutions of long chain polymers*. The investigation of the viscosity and other properties under conditions of continuous rate of shear has been extensive, especially i n connection with their use as incendiary fuels, but the nature of their physical properties under conditions of small strain i s not as well known* Since the investigation of the visco-elastic behavior with small strains has given useful information about the structure of linear polymers and their solutionsland of three dlmensionally cross linked elastomers 2 i t seems desirable to carry out investigations of this type for aluminum soap hydrocarbon gels* The relations between visco-elastic behavior and molecular structure i s understood at least qualitatively for such systems^ so that i t can be expected that some information about the actual structure can be concluded from the study of mechanical behavior* Most of the information about the structure of aluminum soaps; i n solution has been obtained from the study of very dilute solutions*.. The investigation of more concentrated solutions by means of mechanical measurements should give additional i n f o r -mation about the structure of the gels* The most extensive study of these systems subjected to small strains i s probably the work done using the Resonance Elastometer by Van Wazer, Goldberg and Sandvik5>6 and by Gunn*? However the results of measurements with the Resonance Elastometer appear to be unreliable since i t i s d i f f i -cult to achieve &< consistent interpretation of the measurements at different frequencies or of measurements with gels of different composition. For example, although the modulus of r i g i d i t y may be expected to remain constant or to increase with increasing frequency, measurements by Quzm7 using the Resonance Elastometer at 1 to 50 cycles per second have yielded values for the dynamic modulus of r i g i d i t y 3 to 5 times higher than the values obtained by the present author using a different method at 100 to 500 cycles per second**. The latte r measurements were made by studying the propagation of transverse sonic waves by the method developed by Ferry9, There seems to be no reason to doubt the results obtained from wave propagation, measure-ments- especially since i t has; been used successfully by Ferry on solutions of linear polymers and found to agree well with results obtained by other methods-*-. This method has the advantage that measurements are made i n the bulk of the material so that possible sources of error such as wall effects, work hardening, etc*, are either not present or may be considered to be negligible. I t has:the disadvantage of being limited to a rather narrow frequency range and of being applicable to only selected gels because of large variations i n the coefficient of strain birefringence.. The present thesis reports on the design of a new; Instrument for the measurement of physical properties; and on the development of methods; of making these measurements over a larger frequency range than has previously been achieved with a single instrument., A. concentric cylinder geometry has been used i n the instrument and the mathematical analysis has been carried out more rigorously than was done for the Resonance Elastometer and similar i i Instruments i n the paat*. Although i t i s probable that some of the results could be obtained more accurately by other methods the emphasis i n the present investigation has been on covering as large a frequency range as possible i n a single instrument at some sacrifice of absolute accuracy» This approach has been especially profitable i n the case of aluminum soap gels since, as w i l l be seen i n the following, i t turned out that the principle relaxation takes place i n a frequency range far removed from the region covered by dynamic measurements* The material given i n this thesis i s presented as follows* A phenomenological theory of visco-elasticity i s outlined, following Ter Haar10,. This theory i s then applied to a mechanical model representing the instrument system used for the measurements*. The mathematical analysis necessary for making measurements and for the interpretation of experi-mental results i s given* The design of the new. instrument, which w i l l be referred to as the Visco-elastometer, and the procedure of measurement i s outlined* AM, f i n a l l y , the results of measurements on a series of gels are given and the significance of these new experimental facts i s discussed i n relation to previous results and also i n relation to possible molecular structure. H i CHAPTER I THEORY Phanomenologlcal Theory of V i g W f r U i t t r t t j A model consisting of a Hookean elas t i c spring i n series with a dashpot with Newtonian Viscosity as a representation of the mechanical behavior of a l i q u i d was postulated by Maxwell 1 1, Figure 1. To represent the more complicated behavior of polymers, polymer solutions, metals, e t c , i t i s necessary to use combinations of Maxwell Elements, or similar elementary models*. One should i n fact use & continuous spectrum of elementary elements but i n practice i t i s usually found that a rel a t i v e l y small number of discrete elements gives a good approximation. At & single frequency or over a small frequency range one or two Maxwell Elements are sufficient* Where the mechanical behavior i s represented by a number of elements of different relaxation times the contribution of each element to the measured value of viscosity and elastic modulus depends on the length of time over which the experiment i s carried out or, i n the case of measurements with alternating stresses, on the frequency* Those elements with relaxation times short compared to the duration of the experiment or to the period of the alternating stress w i l l behave mainly as viscous elements, those with relatively long relaxation times w i l l appear as simple e l a s t i c elements* For this reason i t i s necessary to measure over as large a range of time or frequency as possible i n order to determine a distribution of elements which w i l l adequately represent the behavior of the particular substance* Flgure /a. S/ny/e Moy we// Element \ \ t \ \ \ figure / b. A/<7jr we// E^/ef7?en ts in Para//e/ ! A i /<? 7^ 0 //<? vv poge /. The general theory of viscoelastic behavior has been considered by Ter Haar 1 0, S i p s 1 2 . Alfrey3 for polymer* and by Zener 1^ i n connection! with anelasticity i n metals. These investigators have been principally concerned with the application of the theory to measurements on solids or glass-like materials such as amorphous polymers, ceramics and metals* With these materials extension tests are usually used and relatively large forces are produced for small displacements* For gels, which w i l l not support their own weight, i t i s necessary to use shear strains and since the modulus of r i g i d i t y i s small the ratio of force to deformation i s much smaller* In principle the general theory applies equally well to both cases but the application of the theory to experimental results i a simpler for extension tests* Consider a deformation, \ , produced by a force, /•"", on a single Maxwell Element, Figure 1 (a), with modulus of r i g i d i t y , , and coefficient of viscosity,/? * The relation between the deformation and the force i a given by where ^ g </A etc** eft I f X i s given as a function of time equation (1) can be solved for /="as a function of time*. This leads to the integro-differential equation 3 where; V i e defined as the relaxation frequency* I f there i s no deformation at time t = 0 X(o) * 0 and /=Yo) • 0 (3) and from (2) and (3) For a number of Maxwell Elements i n parallel i s the same for a l l and the forces add so the generalization of equation (4) i s / r * ^ I < ? / e * ^ ~ ° A « r > c ^ / ( 5 ) * o To generalize farther to a continuous spectrum of relaxation frequencies l e t with the normalization condition 00 00 o o be the fraction of elements with between Q and £?plu» </G and y) between V and V plus * The fraction of elements between y and, V plus di> i s given by 00 o A better representation of the relaxation spectrum i s given by oo (6) This gives a better description of the mechanical behavior because i t Includes the influence of the modulus i n the distribution. Those elements with the highest contribution to the t o t a l modulus w i l l always predominate i n determining the viscoelastic behavior. From equation (6) the average modulus of r i g i d i t y and viscosity are given by eo oo O O ^ ° The generalization of equation. (4) to a continuous distribution of relaxation times i s given by o o (8) O o A third function ^ a ^ e a n be defined such that CO y(*> = / - _/ f/Vcy) eV *</» (9) o where <$ = Xc. 5 Since //(V) i s a non-negative function i t follows that ^ teXa a mono-tonouely decreasing function for increasing time, Also from equation (7) i t follows that fl/o) * 0, Using (9) equation (8) can be written i n the form pft:-*H c/zr ( 1 0 ) o I f a deformation ^ & i s applied i n f i n i t e l y fast at time t = 0 and from equation (10) /=?<» = x.C-So pfc> i s the Relaxation Function, I f , as a f i r s t approximation, i t can be assumed that & i s independent of A , equation (10) can be written o (U) where Ter Haar 1 0 has discussed equation (11) i n d e t a i l , showing that i t leads to a second equation relating the relaxation and creep functions. He also discussed the application of the theory to various possible experimental 6 methods; stress relaxation, creep, extension at constant rate, etc.. He has shown that the best method of finding the function A^yJup to ~ /aa c/s i s by relaxation experiments. Above this frequency vibrational measurements are better. In a stress relaxation experiment i t i s assumed that a known deformation i s suddenly applied and then held constant while -foe force required to produce and maintain this deformation i s recorded. With solids this i s a valid and practical assumption. The ratio of stress to strain i s large and experimental methods are available for measuring these large forces and small displacements accurately. For a weak gel the ratio of stress to strain i s no longer relatively large. In general the methods of measuring stress involve displacements of the same order of magnitude as the deformation of the gel. Similarly methods of measuring displacement generally involve relatively large forces. Without very elaborate experimental methods i t i s not possible to j u s t i f y the assumptions of a simple relaxation or creep test. The experimental arrangement developed i n the present investiga-tion for testing gels i s shown i n Figure 2. A pair of concentric cylinders are used, the inner i s suspended on torsion wires and the outer can be rotated about the common major axis. I f the annular space between the cylinders i s f i l l e d with a gel and the outer cylinder suddenly rotated through a small angle, the inner cylinder immediately rotates through some smaller angle and then returns to i t s equilibrium position as the stress of the gel relaxes. I f the torsional stiffness of the suspension i s known, a record of the angular deflection of the inner cylinder with respect to time upper torsion wire 'nher cy/inc/er outer- cy/inder-9e/ Vfo'y?. /7 ure Z Test Ce// to f*o//ow poye 6 7 gives; the carve fY£) vs. time* The difference between the fixed deflec-tion of the outer cylinder and the deflection of the inner cylinder gives \C£) vs. time.. I f these curves can be approximated by analytic expressions A ^ c a n be found and equation (11) can be solved. This method of finding the relaxation spectrum has; been used i n contrast to Ter Haar's method of using a delta function.approximation to obtain from an ideal experimental relaxation curve* Rewriting equation (11) t This i s the well known Faltung Equation and from the theory of the Laplace Transform and yet) - L~'\ Ifxw- Atfj (12) (13) L[Xt±) =l{Xc*)]/JfH&] CU) (15) The upper l i m i t off 1> involved i n an experiment of the type described above i s fixed by the f i n i t e time required to make the i n i t i a l a deflection of the outer cylinder and by the response time of the inner cylinder system to stress. M l elements with relaxation times shorter than the longest of these tvo factors w i l l relax before maximum deflection of the inner cylinder i s reached and therefore their effect w i l l not be observed,. To measure the effect of the elements with, relaxation times below the time required to make the deflection of the outer cylinder vibration measurements must be used* In.this case the outer cylinder i s held fixed and the inner cylinder made to oscillate sinusoidally about the longitudinal axis,. By a suitable technique a dynamic modulus of r i g i d i t y and coefficient of viscosity can be found for each frequency. In general both these quantities are frequency dependent characteristics of a Voigt Element, i , e , spring and dashpot i n p a r a l l e l . The Determination <a fft&mAAfokKtm ^\m\^m Measurements The mechanical model representing the experimental arrangement described i n the l a s t section i s shown i n Figure 3, The torsion wires are represented by an elas t i c spring of modulus, S, and the viscoelastic material i n the annnlar space by a para l l e l array of Maxwell Elements, The relaxa-tion experiment i s equivalent to suddenly displacing the bottom terminal by a known amount and recording the subsequent displacement of the junction between the spring and Maxwell Elements, I t i s assumed that the angular displacement of the inner cylinder can be approximated by the sum of a f i n i t e number of exponential terms. \ \ \ \ \ \ \ s > em G n Me eh a n/c <?/ Mo c/e / for Re/oxat/ofn Me <?sur-eme/its to fo//ow f>oye I f >€» = O , ( 1 6 ) i s the angular displacement of the outer cylinder, and € J t >o (17) the angular displacement of the inner cylinder* and * - I ^ then *^ a j t In this equation i s the torsional r i g i d i t y of the suspension and <P i s the sum of the modulii of r i g i d i t y of the gel each multiplied by the c e l l constant, which w i l l be discussed later. From the above assumptions * W (18) y ***** r \(*> = ) Ste) / 2 _ e r,J2?aK ****** x From equations; (16) to (19) equation (12) can be written i n the form S ~~~*r~ " / / >» /H ' # 0 f tv Q. A-l / tt-r (20/ This i s a Faltung Equation and taking the Laplace Transform of both sides gives max and the inverse transform gives /7 11 flf£)caxk be found from this equation by the method of residues. /VW)can be determined since £ (/~ fm>) i s the Laplace transform of AtfJ, from equation (19)* Obviously the above procedure results i n a fpfe) with the same number of exponential terms as were originally used to approximate the experimental curve.. I t w i l l not result i n a unique set of terms for yfif} since the set used to approximate the experimental curve w i l l depend on the method of curve f i t t i n g . A simplification of Irony's Method^ has been used for curve f i t t i n g . This f i t s two exponentials to four points; at equal increments i n time (Appendix: I ) . The l a s t decade i s f i t t e d f i r s t , then the two resulting exponentials are subtracted from the rest of the curve. A decade near the end of the difference curve i s f i t t e d , the exponentials subtracted again, and so on u n t i l the beginning of the experiment i s reached. The experimental curve can be f i t t e d to better than experimental accuracy by this method so that the resulting should give a good indication at least of the location and approximate height of the peak i n the relaxation spectrum. —M<*) from Vibration Measurement^, Vibration measurements must be used at frequencies above the region covered by the method of the preceding section. Any viscoelastic material can be represented at a single frequency by a Voigt Element with frequency dependent components, £'(<+>)andr)'(w)+ This i s equivalent to a 12 complex modulus of r i g i d i t y . I f the applied force i s given by and the resulting deformation i s given by (24) A r / ; = A e^"* (25) o then - (r(cvl - &Sou) J t*> p '(ou) ^ 6 ) Measurements are made under steady state conditions with /^-oo)- \(-oo) - O so equation (8) should be written t =•J</» At»J e Mtrj </?r ( 2 7 ) O —oo Substituting from equations (24) and (25) O o —oo 00 Therefore and from equation (26) o (28) I f Gff> and^^were known for o c*s - o& , /Vf*>) could be found by making a Fourier and then a Laplace transform* However* this complete range i n oo i s never covered experimentally so approxi-mation methods must be used to determine from the measured values of 6"feu) and y'fou). Equation (28) for £ro) can be written The function //(X + ty^ ) has a maximum at X - / and approaches zero for X-±o and X-^ce> *. I f a - function i s taken as an approximation for this function* equation (28) becomes OO C'cov) ~ S / C L U X ) $(X_N < / K (29) and /\/foo) ^ <?fcu) (3Q) Alternatively equation (23) can be written Gr~> « J^AV(M) ~ (31) The function /yfrx*-*-!) behaves l i k e a cut-off function, approximating 1 for OsCX4./ and 0) for X> / Using this approximation equation! (31) can be written cu 80 Similar approximations can be made using equation (28) for yttu), giving V7. OU Theory of the Concentric Cylinder Yiseo-Slastometer The same experimental arrangement was used for making vibration measurements as was used for relaxation measurements^ Figure 2. The inner cylinder i s made to oscillate sinusoidally about i t s major axis by passing an alternating current through the c o i l of an electro-mechanical transducer. 15 The motion of the cylinder i s opposed by the suspension wire and by the gel i n the annulnr space between the cylinders* I f the equations of motion of the system are known and can be solved the modulus of r i g i d i t y (P ' and coefficient of viscosity y r can be found for each frequency from the load on the cylinder due to the gel. I t i s easy to set up the equation of motion and solve i t rigorously but the solution i s of such a form (equation (40)) that the r i g i d i t y and viscosity cannot be obtained without making an approximation. One approximation made to simplify the mathematical analysis has been to assume that the effect of the i n e r t i a of the sample i s negligible*5»° This assumption would lead to large errors with the Visco-elas tome ter. With this instrument A^/R^s small and i t was found that reasonable accuracy could be achieved by assuming a plane coordinate system instead of cylindrical and including the effect of in e r t i a . This avoids the appearance of fiessel functions i n the solution* However Markowitz^5 has recently developed a method of reducing the equation i n Bessel functions to a series i n algebraic terms* This method has been used for the Visco-elastometer i n the following manner. To get the equation of motion, l e t t] = the radius of the inner cylinder /£ = the radius: of the outer cylinder h s the depth of immersion of the inner cylinder i n the gel &Crt)~ the angular displacement at radius r f = the density of the material & =» the modulus of r i g i d i t y of the gel >?' = the coefficient of viscosity of the gel 16 The equation describing the motioniof a cylindrical lamina of height, /t and r a d i i r and r + c/r ±9 2w+7A</+/>£e =r ( ? i f 2 ^ J i e ) ^ < A or (34) Substituting, for the case of sinusoidal motion, f^r^e^00^^ ecrt) (35) gives +2.24) +v'C?* +J-H) which reduces to 414 +J~?L$.+ ">*~ 0 = o (36) r f Now putting puts equation. (36) i n the form 17 With = a2' this becomes which i s Bessel's Equation of order one with a solution; For an oscillating inner cylinder and fixed outer cylinder the boundary conditions are 0(*7) - 0, 0<T3.)-= 6 Therefore A + « rrfi giving A - *7 0, X(a1) The torque acting on the oscillating inner cylinder due to the gel i n the annular space i s given by (39) 18 From the recurrence relations </_ <^(af~)-= a etc. «//-where <J£(Z) - </<%(z) ^ fay-fey ftp*;)) - JL\ (40) To obtain the values of <? and y ', from the measured torque, equatiom(39) must be solved for these quantities after substituting for ° ^ % h from equation ( 4 0 ) . The solution of the resulting equation i s made d i f f i c u l t by the fact that a l l the fiessel Functions have complex arguments involving the unknowns* Until recently this d i f f i c u l t y had not been overcome successfully and this lead to the various assumptions discussed above. However the expression 40 for </^^ can be reduced to a more con-venient form using Markowitz1-1-^ results (41) fO 19 The constants 6 and 6 are given 2-j +1 n*/; 1J -hi i n Markowitz;1 paper 1^. How writing V JL A - # m tt*o which can also be evaluated using the Markowits results, equation (41) can be reduced to 1 ' ^ / / (42) Substituting from equation (42) into equation (39) and (40) with known and using a. * / \ ^  gives an equation which can be solved for £ ' and 7 to any desired degree of accuracy, depending on the number of 20 terms used i n expansion (42)« Theory of the E l e n t r r " - ^ f t ^ ^ ^  i T h i M * " * " ' To make use of the equations developed i n the l a s t section for determining the dynamic modulus of r i g i d i t y and coefficient of viscosity i t i s necessary to measure the torque exerted by the test material on the oscillating inner cylinder,, There are several methods i n use for doing this. Some involve e l e c t r i c a l or mechanical drives for the determination of resonance points of the gel-instrument system^* 1 0 . Such resonance methods are not particularly accurate, permit measurements only at di s -crete intervals: and usually involve rather large deformations of the material tested. Other methods use electro-magnetic driving and detecting coils mounted r i g i d l y on the moving element. The mechanical impedance presented by the gel i s calculated from the relative amplitude and phase of the driving and generated voltages. In the present vork an electro-mechanical transducer method has been used. The method uses only a single c o l l . mounted on the moving element, the impedance of the c o i l being measured by an alternating current bridge. With the bridge method high sensitivity i s possible with small driving voltage on the c o i l and therefore the displacements can be kept reasonably small. Further, measurements are,not limited to points of resonance but can be made over a continuous range on either side of the mechanical resonance of the system. This also results i n smaller deformation of the test material. The theory of transducers has been discussed by Mason1''' and the method has been applied successfully to the 21 measurement of polymers by Ferry and others*. The difference between these methods and the one used here w i l l be discussed later* In the present series of measurements i t was necessary that the cylinder be driven with a rotational o s c i l l a t i o n about i t s longitudinal axis* For this purpose the c o i l was mounted r i g i d l y on the inner cylinder and i n the magnetic f i e l d of shaped pole pieces* The c o i l and poles were made i n such a way that i t could be assumed that the magnetic f i e l d was radial* The motion of the cylinder may be analyzed as follows* Consider a rectangular c o i l i n a uniform magnetic f i e l d , the c o i l being suspended so that i t s motion i s restricted to rotation about i t s long axis* Let; & 8 f i e l d strength i n gauss 4 = length of c o i l normal to the f i e l d f » radial distance from the centre l i n e of the c o l l to the conductor /- = inductance of the c o i l i n henrys ^ = resistance i n ohms J a moment of Inertia of the c o i l and associated parts i n gram centimeter 2 5 - torsional r i g i d i t y of the suspension i n dyne centimeters/radian 6 m current through the c o i l i n amperes voltage on the c o i l i n volts torque i n dyne centimeters & - angular displacement of the c o i l i n radians When the annular space i s f i l l e d with a i r only the torque on the c o i l due to the current and the displacement of the c o i l i s r*j(u>T- §_) e - Off**-) /c'; = o ( 4 3 ) CO and the voltage i s I f 21 o~ the impedance of the c o i l when clamped ^ = the impedance of the c o i l free to move i n a i r ^m-1/ (usj~- S_) s the mechanical impedance of the system, then i n a i r ~a C4J and or Z„ . (gVA) U 5 ) 23 Taking only imaginary parts this can be written cT _ _S_ _/ = X- Xo  This equation permits calibration of the instrument for i f the right hand side i s plotted against f c r different values of tf~ a series of straight lines i s obtained* and >y =» the slope (47) c7/ « the intercepts with the axis The set of equations (4.7) can then be solved for J * » c7~> /O • When the measurements are made with a gel between the cylinders i t s effect i s added to Z-m i n equation (45) using the results of the preceding section* The resulting equation can then be solved f o r ^ f ^ e o ) and j?^*<x>) « 24 CHAPTER II APPARATUS Since liquids are unable to withstand tensile stresses of any magnitude or duration the only type of stress which may be applied to (polymer) solutions i s a shear stress and the fundamental problem i n designing an instrument for testing such solutions l i e s i n the choice of method for applying the stress*. The geometry of the system must be such that stress i s as simple as possible* The simplest type of motion involving pure shear i s plane laminar motion but since this i s not easily achieved experimentally i t i s usual to approximate i t by stressing between- con-centric cylinders* To have a good approximation to plane laminar motion the ratio of width of annular gap to radius must be small* Since stress-strain relationships may be expected to be nonlinear i t i s preferable to use strains as small as possible and also to make a l l measurements i n a single test c e l l * By using the same geometry and size of sample over the whole frequency range one can assume that errors due to non-linear effects w i l l be approximately constant throughout* For this reason;the present instrument was designed so that strains; could be produced either by os c i l l a t i o n of the inner cylinder or by rotation of the outer cylinder through a small angle, i n both cases relative motion i s about the common longitudinal axis* The test c e l l consisted of an outer cylinder of precision bore glass tubing of inside diamter 1*000 centimeters plus/minus 0.0005 centi-meters, and outside diameter 1*270 centimeters plus/minus 0*0025 25 centimeters. The inside surface was ground and polished and the outer surface was ground concentric with the inner surface to the above tolerances. A glass outer cylinder was used to aid i n alignment of the cylinders during assembly and also to permit visual inspection of the gel for a i r bubbles, etc. during a measurement.. The portion of the inner cylinder i n contact with the gel was thin walled aluminum tubing with an . outside diameter of 0.900 centimeters; plus/minus 0..0025 centimeters. This gave an annular gap 0«,05 centimeters wide. Since the moment of i n e r t i a varies as the fourth power of the radius the diameter of the inner cylinder i s determined principally by the mechanical resonant frequency required and the stiffness, of the suspension used. To give adequate sensitivity the dimensionsof the apparatus were chosen so that the contribution of the gel to the r i g i d i t y of the system was of the same order as the stiffness of the suspension wires for a resonant frequency of about 125 cycles per second with no gel i n the instrument. The value of 0.05 centimeters for the width of the gap was a r b i t r a r i l y chosen as a practical l i m i t for this dimension. A gap of this size gives good sensitivity and at the same time allows reasonable tolerances on alignment and concentricity of the cylinders, while a ratio of outer radius to gap of 10tl ensures approximately laminar strain. Over the range of r i g i d i t i e s expected the wave length of transverse sound waves varies from 1.5 to 3.5 millimeters at 250 cycles per second. This w i l l be the approximate upper li m i t of frequency since the gap must be considerably less than a quarter wavelength. 26 A schematic drawing of the elasto-viscomater i s shown i n Figure 4* The support for a l l components i s an open rectangular box, with two sides, top and back of three eighths aluminum plate, and the bottom of one inch thick brass plate. The outer glass cylinder i s held by a collet chuck mounted i n b a l l bearings. The inner cylinder i s suspended from either end by music wire held at the bottom and positioned at the top by small, three jaw chucks. To eliminate errors introduced by f r i c t i o n where the lower torsion wire passes through the bottom of the c e l l the length of the cylinders and lower torsion wire are fixed so that the wire i s clamped by the chuck just below the point where i t passes through the brass cap which forms the bottom of the outer cylinder.. A small f e l t pad between the chuck and brass cap prevents leakage out of the hole for the wire. Tension i s applied to the suspension by a pair of threaded sleeves at the top of the apparatus before the upper wire i s clamped i n the chuck. The bottom chuck i s set i n a Morse Tapered plug so that the whole c e l l assembly can be removed through the bottom plate for cleaning. Dynamic measurements of r i g i d i t y and viscosity are made by a single c o i l transducer method. A detail drawing of the inner cylinder i s shown i n Figure 5. The c o l l consists of about two thousand turns of Number 44 B&S gauge copper wire wound onto a powdered iron core. This i s then set i n a slot milled i n a solid bakelite cylinder and tight l y clamped by four 2-56 screws. The magnet pole faces c o i l and core are a l l sections of cylinders so that for the small displacements involved the c o i l may be assumed to be moving i n a constant magnetic f i e l d . To make measurements with the c o i l fixed the inner cylinder i s clamped by a small spring loaded plate which bears on the top. The magnetic f i e l d i s produced by an Alnico Torsion Wire 27 Magnet and mild steel pole pieces. The e l e c t r i c a l c i r c u i t for making dynamic measurements i s shown i n Figure 6. A Maxwell Bridge c i r c u i t i s used with a Hewlett Packard Type 202-B Oscillator as source and a Hewlett Packard Type 400-6 Vacuum Tube Voltmeter as detector* The bridge i s made up of a General Radio Decade resistance box with 0*1, 1, and 10 and 100 ohm steps. General Radio Type 500-ff fixed resistors, a General Radio Decade Condenser box with 0.001, 0.01, 0.1, and 1 microfarad steps and a 100 to 1100 micro* microfarad continuously variable General Radio Air Capacitor. To simplify calculations the arms of the bridge are arranged so that the difference between the resistance with the c o i l clamped and i n motion gives the change i n resistive component due to mechanical resistance directly. Similarly the change i n capacity C i n microfarads gives the change i n mechanical inductance directly i n henries. With this c i r c u i t changes of less than 0.1 ohm resistance and 0.01 ohm inductive reactance are easily detectable. Relaxation measurements are made by rotating the outer cylinder through a small.angle and recording the angular motion of the inner cylinder on moving f i l m with a li g h t source and optical lever. A small front sur-faced mirror approximately 5 millimeters square i s mounted by means of a s p l i t collar which i s screwed onto a short threaded boss integral and concentric with the top of the inner cylinder. A Genco 2 watt zirconium arc and a one meter focal length lens are used as a light source. The light path i s doubled once by a plane front surfaced mirror and the source focussed on the f i l m of an oscillographic recording camera.. Since the suspended mirror must be remounted for each Osc/ / / otor /OOO rx. Voffmerer- ALc= AC Sri dye C/rco/t to -fo//ow foje 27. 28 measurement the second mirror was made moveable l a t e r a l l y and about horizontal and vert i c a l axes to position the l i g h t spot on the film. The camera used was a Gossor Model 1428 Oscillograph Camera with nominal continuous: f i l m speeds of 0.05 to 25 Inches per second. The camera lens was removed and the li g h t source focussed dire c t l y on the 35 mm fi l m . The l i g h t path from mirror to f i l m i s four meters so that the maximum measurable deflection of the inner cylinder i s approximately 0.003 radians corresponding to a displacement of 2.4- centimeters i n the plane of the film. The top plate of the bearing and collet assembly holding the outer cylinder carries a r i g i d arm which i s pulled up to a fixed stop on the frame of the instrument, Plate I, by a strong spring. Accurate and reproduceable small angular deflections of the outer cylinder can be produced by withdrawing a short spacer of music wire of known diameter from between the arm and the stop a fixed distance from the axis of the instrument. The viscosity of an aluminum soap hydrocarbon gel at low rates of shear i s of the order of 10^ poise and gives r i s e to considerable d i f f i c u l t y i n f i l l i n g a c e l l of small dimensions since the c e l l must be completely f i l l e d without a i r bubbles and without disturbing the relative position of the concentric cylinders, Even with a tension of several kilograms on the suspension the relaxation time and r i g i d i t y are so large that even small l a t e r a l displacements of the inner cylinder are not corrected i n a practical time interval after f i l l i n g . I t i s not possible to f i l l the outer c e l l and then place the inner cylinder i n position accurately. The best method found was to assemble the apparatus completely 29 and then f i l l the c e l l from the bottom of the outer cylinder by forcing the gel through a aide tube i n the brass bottom cap. Figure 4* A brass syringe was used to force the gel into the c e l l * The cylinder of the syringe was f i r s t f i l l e d by inserting one end into a jar of gel with pro-vision for sealing the top of the jar by clamping a thin brass disc, attached to the cylinder, onto a rubber gasket. Air pressure was then applied to the top of the gel through a small i n l e t tube and the gel vas allowed to r i s e slowly u n t i l i t completely f i l l e d the cylinder* The cylinder was then removed, the syringe assembled, threaded onto the side i n l e t at the bottom of the c e l l and the gel forced into the c e l l by moving the plunger by hand* When measurements were f i r s t made using gels i t was found that the loss of solvent at the top of the c e l l was so rapid that a skin of almost pure soap formed between the cylinders* The mechanical effects of this skin completely overshadowed the effects of the gel i n the annular space between the cylinders* I t was also found that surrounding the top of the outer cylinder by an annular ring of pure solvent was not sufficient to eliminate this effect completely* I t was found necessary to add a brass top to the glass tubing. Figure 4* This was i n the form of a shallow cup which f i t t e d t i g h t l y where i t overlapped the glass cylinder and then Increased to an inside diameter of one inch above the top of the glass. The c e l l was f i l l e d u n t i l the gel rose above the top of the glass to form a ring about two millimeters thick around the top of the c e l l * The cup was then f i l l e d with pure solvent* The amount of gel i n the ring around the top was sufficient so that the solvent did not penetrate far enough during a run to dilute the gel i n the annular space* 30 CHAPTER III CALIBRATION OF THE VISCO-ELASTQMETER Measurement off thft Mechanical Constants S r J and, ( B l . r ) ^ " 9 The Torsional stiffness of the suspension vires, S, moment of inertia of the inner cylinder and c o i l assembly, J, and the transfer constant ( B l r ) 2 l 0 ~ 9 were a l l determined from bridge measurements using equations (46) and (47)• The resistance and inductance of the c o i l were measured with the c o i l clamped and free to more i n a i r at equal intervals of xfi/cts 2 on either side of the mechanical resonance point. This was done for the cylinder alone and for the cylinder loaded with added discs: of known moments of iner t i a . The moment of in e r t i a of each disc was calculated from i t s measured mass and dimensions* The procedure used was to measure both the clamped and free impedance at each frequency. I t was not possible to use a master plot of the clamped resistance and inductance against frequency for although both these quantities are only s l i g h t l y frequency dependent the variation i n impedance with temperature i s so large that i t causes excessive errors i n the measured values of mechanical resistance and inductance* The resistance of the c o i l was about 625 ohms and the temperature coefficient of resistance for copper i s approximately O.OOA/^ C so that a temperature difference of one degree caused a change of 2*5 ohms. For this reason changes of a small fraction of a degree i n ambient temperature can cause appreciable errors i n both the real and imaginary parts of the calculated mechanical impedance* 31 To determine the mechanical constants the quantity J_ Xo-X was calculated from the bridge measurements and plotted against /0%f$ar each moment of inertia* This gave a set of paral l e l straight lines* The equations obtained by substituting the values of the intercepts of these lines into equation (4-7) were solved for the moment of in e r t i a of the suspension, J, and the constant (Blr) 210"9. The common slope of the lines gave the torsional r i g i d i t y , S;, when (Blr) 210**9is known* In i t s f i n a l form the instrument had the constants (ELr)^10"9 «= i # 0 7 ± 10* ohm dyne second centimeters SI = 8*6 x 10-* dyne centimeters/radian J = 1,66 gram centimeters 2 Calculation y d B ^ a ^ iMwflal, Verification of the C e l l Constant The dimensions of the concentric cylinder c e l l were; inside diameter, r ^ - 0*45 centimeters outside diameter, r% = 0.50 centimeters; depth of immersion of the inner cylinder i n the c e l l , h = 7*1 centimeters Substitution of these values i n equations; (42) and (40) using Markowitz1 method outlined i n the section on theory gives 32 From equation (39) the torque acting on the inner cylinder due to the gel i s Using the f i r s t three terms of equation (49) as a f i r s t approximation, equation (45) gives « (50) ^ J U v f - / f o ) 2 ^Cr-A-o)'" j (51) **** f + Q r ~ x . ) z ) m 33 Equations (51) and (52)) were checked using a Newtonian liquid. The viscosity of a glycerol water solution was determined using both am Ostwald Viscometer and the Visco-elastometer. The Qstwald Viscometer was calibrated with water at room temperature. Measurements with the Visco-elastometer were made over the frequency range 30 to 250 cycles per second and no systematic variation with frequency was found. The averaged values of viscosity by the two methods aret 6,35 $ *5 poise (Ostwald) 6,233 - ,45 poise (Visco-elastometer) Since these results agree well the constants shown in equation (52) were used in a l l calculations to follow. As a check on the density term in equation (51) the quantity was plotted against A7/to1- on the same graph used for the results of measurements in air. Figure 7, The two plots were parallel indicating a change in the effective moment of inertia of the system as predicted by equation (5l)» The magnitude of the shift in moment of inertia is not large enough to give a reliable check on the factor 5.52 x 1QT2 in equation (5l)« The value of this factor was verified by checking the consistency of measurements;of the rigidity of gels with different moments of inertia added to the cylinder. Since the effect of inertia of the sample is small, i t is only appreciable in measurements at high frequencies where 3A the moment of ine r t i a of the suspension i s small. Since the values of r i g i d i t y obtained using the unloaded cylinder only form a continuous curve with values obtained at lover frequencies and high moments of iner t i a i t vas assumed that the density correction i s sufficiently accurate* 35 CHAPTER IV PROCEDURE The method of f i l l i n g the c e l l has been described i n the section on apparatus. After f i l l i n g , the gel was allowed to relax for a period of th i r t y minutes. Relaxation methods were always made f i r s t . The small mirror was mounted on the suspension and the l i g h t spot located on the ground glass screen of the camera before the instrument was f i l l e d . The alignment of the cylinders could then be checked after f i l l i n g by noting the amount by which the light spot was displaced on the screen after relaxation was complete. There was always some displacement but never more than the equivalent of a quarter of a degree of the mirror. The change i n relative position of the cylinders i s negligible for an angular displacement of this magnitude. Two relaxation measurements were taken; the f i r s t for about five seconds at a f i l m speed of 3.57 inches per second and the second for five minutes at a f i l m speed of 0.0714 inches: per second. The gel was allowed to relax for a period of five minutes between these two recordings. In the majority of cases the angular deflection of the outer cylinder was chosen so that the maximum deflection of the l i g h t spot was about twenty millimeters. For very weak gels this i s not possible without using excessively large strains so smaller deflections had to be accepted. The deflection of the li g h t spot was measured directly off the 35-mm film, using a Baush and Lomb Spectrum Measuring Magnifier. Readings could be made to the nearest 0.1 millimeter corresponding to an angular 36 deflection of 1.25 x 10"*-> radians. These measurements were made at points corresponding to suitably spaced intervals of the logarithm of the time and are then plotted against the logarithm of time. Dynamic measurements were made on the same sample, readings being taken at ten equal logarithmic intervals over the decade 25.2 to 252 cycles per second. Three different moments of i n e r t i a were used to cover this frequency range. Small brass discs were used to change the moment of in e r t i a of the system by screwing onto the threaded boss on the top on the inner cylinder. When not i n use the discs were held against the top chuck with the suspension wire passing through the central hole. With this method of mounting the moment of in e r t i a can be changed by adding successive discs of known moment of inertia without disturbing the apparatus. The mechanical constants of the apparatus and frequencies at which measurements were made are shown i n Table I. 37 Table I Mechanical Constants of the Visco-elastometer Moment of Inertia gm.cm.^  Torsional Rigidity dyne cm./rad« Mechanical Resonant Frequency cycles/sec. Frequency of Measurement cycles/second 1,66 4.82 21.52 8.6x10-8.6x10* 8.6xl0 5 114.5 67.2 31.8 252 200 159 126 100.5 100.5 79*7 64.5 50.3 50.3 39.8 31.75 25.2 Dynamic measurements vere a l l made with an applied voltage of 4*5 volts rms. on the bridge. This gave a current of approximately 2.25 milliamperes through the transducer c o i l . This i s near the maximum allowable current at 1500 circular mils per ampere. The reactance of the c o i l i s small compared to the resistance and since the balance was always made with constant resistance i n each arm the e l e c t r i c a l conditions are nearly constant and the bridge always used at maximum sensitivity. 38 CHAPTER V EXPERIMENTAL RESULTS The Effect of Magnitude of the Strain I t i s v e i l known from viscosity measurements that the coefficient of viscosity of polymer solutions decreases: rapidly with increasing rates of shear* The moduli! of e l a s t i c i t y and r i g i d i t y are also known to vary with strain for large deformations* To interpret the experimental results i t must be assumed that Hooke's law holds for the ela s t i c elements and that viscous elements are Newtonian* This assumption was verified experimentally. To check the effect of the magnitude of the strain i n relaxa-tion measurements, records were taken at a f i l m speed of 0*0714 inches per second with different angular deflections of the outer cylinder* There was no appreciable change i n the modulus of r i g i d i t y or i n the shape of the relaxation curve when the deflection was doubled* A better check on the effect of strain and rate of strain was made from the transducer measurements. Equations (43) and (44) can be solved for the angular deflection <^r*> of the inner cylinder, giving 39 So that the maximum angular displacement of the inner cylinder i s W a x "—" and from equation (45) >n<*x ° For i =^2.25 x 10" 3 amperes and (Blr) 210" 9 «• 1.07 x loA Assuming laminar motion as a f i r s t approximation the strain i s given by »*a.K (54) C O AS an example |Z**^q ' had max!mum and minimum values of 8.61 x 10~3 and 3.34 x 10T^ corresponding to strains of 0.78 and 0.03 for a l$> solution of Digel i n benzene. The value of the modulus of r i g i d i t y was within experimental error for both these strain amplitudes. Since measurements are made on either side of three mechanical resonant, points for each gel a number of these points of mayttmiin and minimum strain are 40 passed. Since no systematic variation of the measured modulus of r i g i d i t y with frequency was found i t was concluded that the elastic constants are independent of strain over the range used i n these experiments. This statement also applies to relaxation measurements i n which the maximum strain applied never exceeds 0.1, The rate of strain also varies over a wide range i n the dynamic measurements so that i t can be assumed that the mechanical constants are independent of rate of strain too since no systematic variation was found. The Modulus of Rigidity from both the Velocity of Transverse Sonic Waves and V1 soo-elas tome ter Measurements The modulus of r i g i d i t y was determined at the same time for two gels of Octal, Octoic a d d i n Gasoline both by the Wave Propagation Method** and by dynamic and relaxation methods using the Visco-elastometer. The two gels used were prepared at the Suffield Experimental Station i n July, 1951 and stored i n tigh t l y sealed glass jars after being tested by the Wave Propagation Method. These mixes were chosen because i t was known from the previous measurements that they had a high coefficient of strain birefringence ensuring reliable measurements of the wavelength of sound. SSJ Secondly both are particularly stable gels, the value of the modulus, £ , from wave propagation having dropped by not more than 10$ during more than 18 months' storage. 4l For the present tests measurements of the wavelength of trans-verse waves were made at eight frequencies between 100 and 4OO cycles per second using the apparatus and procedure described elsewhere**.. No dis-persion i n the velocity of propagation was found over this range so the average velocity was used to calculate the "wave r i g i d i t y " G- • Dynamic measurements of the modulus of r i g i d i t y and viscosity were made i n the Viseo-elastometer by the transducer method over the frequency range 25.2 to 252 cycles per second* Again there was no dis-persion with frequency and so the average of values found at ten equally spaced logarithmic intervals was taken for the value of the dynamic modulus. G'« The fact that there i s no dispersion i n the value of the modulus of r i g i d i t y with respect to frequency indicates that there i s no appreciable elas t i c effect with a relaxation time within at least a decade below the range of angular frequencies covered by the two methods of dynamic measurements* Relaxation measurements were also made on the same samples and the resulting experimental curves f i t t e d by a sum of exponential terms by the methods outlined previously i n Chapter I* The sum of the moduli! i s taken as the total modulus of r i g i d i t y of the gel* The results of the above measurements of r i g i d i t y are summarized i n Table I I . Table II Modulus of Rigidity of Octal and Octoic Add i n Gasoline (in dynes/centimeters/radian) from Three Experimental Methods Octal Concentration (weight) 5* 6% Octoic Concentration (weight) 2.5% % G (wave propagation) 1U5 2580 G1 (transducer) 1160 2590 G (relaxation) 1280 2730 The experimental relaxation curve for the 6% Octal gel was analyzed using the Laplace Transform method. The mechanical behavior can be represented by the relaxation function No attempt has been made to calculate the dynamic moduli!. G and G1, for the distribution of Maxwell elements represented by the above relaxation function. However the frequency range covered i n dynamic measurements: i s well above the region where any dispersion may be expected except possibly for the f i r s t term. Ferry, Sawyer and AshworthkS have given curves:showing the variation of G/G and G'/G with frequency for a single Maxwell element. These quantities have negligible values at frequences far below, the relaxation frequency and increase monotoni cally as the f r e -quency i s increased approaching unity asymptotically. For both cases the ratio i s approximately one at ten times: the relaxation frequency. The experimental curve for the 5% Octal gel has not been analyzed but i t i s 43 similar i n shape to that of the 6% gel but with the region of maximum relaxation about a decade lower i n time. This i s s t i l l s u f f iciently far removed from the region of dynamic measurements for no dispersion to be expected. Since no dispersion i s expected i n either case direct com-parison of the modulus as found by the three methods i s possible. The, Dependence, of Bjg|dAty on Concentration, Digel Soap solutions i n benzene of weight concentration l&, 5 * 3 * and 6.5$ have been tested i n the Visco-elastometer. The solutions were prepared by slowly adding the soap powder to benzene at 35 degrees centigrade. The solution was stirred continuously with an ele c t r i c s t i r r e r during the addition of the soap and for a period of t h i r t y minutes after a l l the soap had been added. The temperature was held at 35 degrees centigrade by a surrounding water bath for the whole mixing time. Mixing was done i n clean glass jars which were then tigh t l y sealed and stored at room temperature. The mixes were stored for two weeks to ensure complete aging then tested and stored for a further four weeks and the tests repeated as a check on aging and s t a b i l i t y . Some variation i s to be expected between the results at two and four weeks due to effects of impurities, etc. on the aging rate and s t a b i l i t y . The results of measurements of r i g i d i t y are summarized i n Table I I I . The results of transducer measurements shown are probably/more reliable than those from relaxation measurements since they are the average of ten 44 measurements at different frequencies while only a single result i s possible from relaxation measurements* Relaxation measurements are more reliable for the second set of values because an improved method of packing was used at the bottom of the c e l l ( f e l t instead of rubber)* The average of the three best values for each concentration has been used i n determining the relation between concentration and r i g i d i t y * Table III The Modulus of Rigidity, G, of Digel i n Benzene i n dynes/cnryradian Age (weeks) Method Concentration: (by weight) 4* 5.356 6*5* 2 Transducer 1550 3350 1 6650 Relaxation. 1800 2950 5900 6 Transducer 1400 3850 7500 Relaxation 1300 ^ 3950 7400 The logarithm of the average r i g i d i t y i s shown plotted against logarithm of the concentration i n Figure 8. The slope of the straight line through the points 3*15 and the intercept with the log c = 0 axis 1*275 giving, approximately, 45 The Rflflxaiaoq Fun^ion, and Vlsoo.sAfr Experimental corves of angular deflection of the inner cylinder are shown plotted against the logarithm of time i f Figure 9.. A l l the records taken give similar deflection vs log, time plots. By inspection i t can be seen that most of the decay takes place over two or three decades of time so that the mechanical behavior may be expected to correspond mainly to that of one or two Maxwell Elements. This i s confirmed by an analysis of the curves f o r ^ # W . The experimental curves for &/&) were f i t t e d with sums of exponential terms and the relaxation function found using the Laplace Transform method. The number of terms necessary to f i t the curves varied from three to fi v e . & ( £ ) and y/Cfi are shown below for two mixes which had been stored for eighteen months» ft OQW Qctojc Acjd j n Benzene &C*)~J-\o.//e +0.*?e> +-o.2,fe /• o.ffyj'e +*.03e I . -2.9.3* ~/*fr- -<?.ff27* ^o.o&/* -o.oo?7*\ = d./22e - c./oYe -0.22/e - <?. Y9& - o.oS~e / 6% Octal 3& Octoic Acid i n Benzene <$>oo / -//'9S* -0.S-2C -0./2S* -0.03* -0.0O0(tr y/£?= /- <?./3?e> - o. 3?<ye - 0. ?2j-<=> - 0.07/e - o.o7<S>e 6.6% >0<=f-/x/0 0.0/ o./ /.o /o /oo /ooo seconds 46 The relaxation functions for Bigel-Benzene solutions after two and six weeks storage are shown i n Table IV. Table IV The Relaxation Function ^ ^ f o r Digel i n Benzene After Two Weeks' Storage 6.5* Dftmfl After Six Weeks' Storage tyf*) - / - 0.0 7/e - ^ . / 2 z e -0. /VS>e - 0.72 &e -.<?.g>22.Tf -0.0£& - 0.OO32 & 6 , 5 * D^eO, - 0.7-eef* ~ 00 OSS' * psf*) = / - 0. 0 ?9 & - 0,727 e 47 Since no dispersion with frequency- was found i n dynamic measure-ments there i s no peak i n the relaxation spectrum i n the range of relaxation frequency from 160 to 1600 seconds" 1. Therefore the Relaxation Function 4^72^ may be taken as the Relaxation Function for relaxation frequencies between 3x10*3 to 1600 seconds" 1 covered by dynamic and relaxation measurements. The Distribution Function ^ v ^ From equation (9) /V&ie the Inverse Laplace transform of where <£- i s the sum of the r i g i d i t i e s of a l l the elements contributing to • Formally 1^ L.\So»}- I therefore e (55) Using equation (55) A f r y can be written down using the results shown i n Tables III and IV giving the functions shown i n Table V below. 48 Table V The Distribution Function for Digel i n Benzene After Two Weeks' Storage My) - /70 $(y-3.VS) -r/J-o S(V- 0. 773)-r2?of()>-o.fn)H/9oS(V-o.ool8) 6.5% Digel AAy>= 3&f(v,-#37)+2fo$(y'-<>./0y) + j-joo JYv- a. 002 *) After Six Weeks' Storage ]& Digel AiX)= S'?S'tty-0.&22) + ?7S'S(y-0.o£&)f-2f&of(y- 0.00 33) 6.5% Digel Since the peak i n the distribution function w i l l be very close to the term having the highest contribution to the r i g i d i t y the most s i g n i f i -cant term i n the above functions i s the l a s t one i n a l l cases. From these results i t can be seen that the relaxation frequency varies very l i t t l e with concentration. I t i s concluded that within experimental error the relaxa-tion frequency i s independent of concentration. This means that the viscosity depends on the concentration i n the same way as the r i g i d i t y . 49 The viscosities calculated from the l a s t terms of the Distribution Func-tions are shown i n Table VI. Table VI The Viscosity Calculated From The Rigidity and Relaxation Frequency Of The Predominant Term i n For Digel-Benzene Solutions (Poise) Age (veeks) Concentration by Weight (percent) 4 5.3 6.5 2 4.25xl©5 2.2xl0 6 6 2.7xl0 5 8.9xl0 5 1.5xl0 6 The valuesof the viscosity shown i n Table VI are of just the same order as would be found by viscosity measurements at very low rates of shear 2 0. From this i t i s concluded that the same molecular phenomenon i s responsible for both resistance to viscous flow under conditions of con-tinuous; shear and for elas t i c and relaxation effects up to a frequency of at least 1600 cycles per second. 50 CHAPTER VI DISOPSSIOH A new instrument, the Visco-elastometer, has been designed for the measurement of mechanical properties of solutions of aluminum soaps i n hydrocarbons, and similar polymer solutions, over an extended frequency range* The Visco-elastometer has advantages over other instruments reported for similar measurements on polymers i n solution* The transducer method has been used by F e r r y 2 1 with a concentric cylinder geometry but the relative motion i s up and down along the longitudinal axis instead of an angular displacement about the axis* P h i l i p p o f f 2 2 used a geometry and motion similar to Ferry's but used a resonance method of measurement instead of the transducer method*. This type of motion does not give a simple flow pattern nor i s i t convenient for the application of relaxation measurements i n the same apparatus* Oldroyd et al l 6 have recently reported a concentric cylinder instrument using a rotational o s c i l l a t i o n but use a mechanical resonance method for making measurements as i n the Resonance Elastometer* As mentioned before a resonance method permits measurement only at discrete intervals of frequency which for given instrument constants depend markedly on the physical properties of the gel under test. Also errors i n the calculation of effects of i n e r t i a of the sample can lead to large errors i n the f i n a l results. The transducer method has the advantage of being applicable over continuous frequency bands and these bands are easily shifted by simply changing the moment of in e r t i a of the system* The Visco-elastometer has the further advantage over the Resonance Elasto-meter and Oldroyd 1s new instrument i n the method of alignment* Alignment 51 Is maintained i n the Visco-elastometer by using a double suspension on the inner cylinder while Oldroyd et a l use a precision " a i r bearing" and a single suspension. The Resonance Elastometer also uses a double suspension but the lower wire i s long and the packing gland i s near the inner cylinder instead of at the point where the wire i s clamped. This introduces a variable and unpredictable error due to f r i c t i o n i n the packing which i s appreciable i n comparison to the effect of the weak gels. Markowitz et a l 2 3 have also reported a concentric cylinder instrument using a rotational oscillation as i n the Visco-elastometer but they have used two coils i n a magnetic f i e l d , one to drive the inner cylinder and a second to record i t s motion. To calculate the torque on the oscillating cylinder i t i s necessary to measure the relative phases and amplitudes of the driving and generated voltages. Since the output of the second c o i l i s made up of the voltage due to the motion, a voltage induced by the current i n the driving c o i l and induced voltages from extraneous sources the sensitivity of the instrument i s limited. With the transducer method a single c o i l i s used and i t s impedance i s measured using an alternating current bridge. The bridge method of measurement i s more sensitive and since a difference method i s used voltages induced i n the c o i l from extraneous sources are automatically balanced out. None of the above instruments have been used for relaxation measurements and therefore they are a l l limited to rather narrow frequency ranges. With the Visco-elastometer both dynamic and relaxation measure-ments have been made, each over the frequency range where i t i s most applicable. This was made possible by the concentric cylinder geometry and method of suspension and by the application of Ter Haar's phenomeno-logical theory of visco-elastic behavior to the interpretation of the 52 experimental data. The Vlsco-elastometer has been used to make measurements over a frequency range of approximately six decades. This: extends the range covered by a single instrument by at least a factor of three. The upper frequency l i m i t i s determined by the wavelength of transverse sonic waves i n the solution being tested. In principle the lower l i m i t can be extended indefinitely. In practice the lover frequency limit i s deter-mined by the precision of the instrument and the method of measurement and by the time of relaxation of the system. In the work covered i n this thesis relaxation measurements were limited to 300 seconds because this time covered the relaxation spectrum i n a l l cases. Ferry et a l 1 have reported on the physical properties of a linear polymer and i t s solutions over a frequency range of ten decades but these results are obtained from actual experimental data covering only three decades of frequency which are extended to represent the mechanical behavior over the ten decades by the method of "Reduced Variables w 2*,, Four experimental methods are used to cover the above mentioned three decades i n frequency*- The method of reduced variables involves certain assumptions which seem to be plausible and consistent results have been obtained i n i t s application i n the case of linear polymer solutions,. These assumptions are essentially that a l l relaxation mechanisms contributing to C' > and y have the same temperature dependence and that a change i n concentration changes a l l relaxation times by the same factor, and that the modulus of each element i n the par a l l e l array of Maxwell Elements varies linearly with concentration. The application of these assumptions to the case of aluminum soap-hydro-carbon gels could not be ju s t i f i e d on the basis of i t s apparent success 53 with materials such as polystyrene solutions alone* The application of the method of reduced variables requires experimental data giving the relation between the mechanical properties and both temperature and con-y centration and the dependence of steady flow viscosity on temperature and concentration. I t i s hoped that the dependence of the physical constants and the distribution function on temperature w i l l be determined i n future work with the Visco-elastometer. When this i s done and data on steady flow viscosity i s available i t w i l l be possible to check the assumptions involved i n the application of the method of reduced variables. I f this method i s applicable to aluminum soap gels the relaxation spectrum w i l l be known over a very large range of frequencies. The fact that the Visco-elastometer gives the distribution spectrum over at least six decades w i l l give a very good check on the above mentioned assumptions. There i s no molecular theory with which possible mechanical properties could be predicted to compare with the experimental data obtained i n the present work. However the experimentally determined behavior agrees qualitatively with what one would expect for a three dimensionally cross-linked structure. The principle contributions to both r i g i d i t y and viscosity must be due to the same molecular mechanism since the variation with concentration i s the same for both. Both vary with a relatively high power of the concentration i n contrast to pub-lished data on solutions of linear polymers which appear to vary linearly with concentration 2^ or with the concentration to a power less than one2->. i t i s known that pure soaps w i l l not form a gel i n a hydro-carbon solvent. The presence of a polar impurity i s necessary for the formation of a gel4. Since an impurity of this type i s necessary i t i s 54 concluded that the actual molecular structure of the soap i s changed during or immediately after solution.. Most of the data on the properties of the Soaps i n solutions has been obtained from measurements on very dilute solutions. From osmosis and l i g h t scattering measurements Sheffer 2^ has concluded that the molecules are long chain polymers of molecular weight 60,000 to 900,000 formed by weak inter-monomer links such as hydrogen bonds. Alexander and Gray4- have concluded from viscosity and streaming birefringence data on dilute solutions that the gel struc-ture consists of intertwined soap micelles, fibrous i n character, which are easily deformed and stretched. I f these fibrous-like micelles are indeed flexible and easily deformed i t i s probable that they consist of relatively few molecules lying more or less parallel to one another so that the ratio of length to width i s large. With fibres of this type i t would be expected that the behavior would be close to that of an ordinary polymer solution. The elastic effect would be due to deformation of the f i b r i l s probably both crystal-like and retarded i n character. The viscous effect would be due simply to rearrangement of the long f i b r i l s after deformation. I t would also be expected that the distribution of relaxa-tion times would be rather wide with the peak or central portion at a frequency of the order of hundreds or thousands of cycles per second as i n solutions of linear polymers 2^. The wide distribution function would be expected because of a large variation i n the length of the micelles. The measured relaxation spectrum i s quite different from that described above. The relaxation function has a very high peak and i s relatively narrow i n frequency range. Both r i g i d i t y and viscosity appear to be due to the same mechanism and the viscosity corresponding to the peak i n the spectrum i s of the same order as that expected from viscosity measurements at very low rates of shear. A narrow distribution may be expected with a three dimensionally cross-linked structure since the variation i n length of the molecules or f i b r i l s between junction points w i l l be less than the total spread i n length i n the uncross-linked case. Very short molecules w i l l not contribute to the three dimensional struc-* ture and very long molecules w i l l tend to have junctions at mid-points along the chain3. The experimental results obtained with the Visco-elastometer can be explained i f a three dimensional structure i s assumed. The major contribution to the r i g i d i t y i s due to deformation of the links between junction points and both steady flow and relaxation viscosity of the main element i s due to the breaking and reforming of actual bonds. Those elements i n the distribution spectrum with higher relaxation frequencies probably represent retarded elastic effects associated with the uncurling of molecular chains. This behavior would be expected both for a three dimensional structure made up of single molecular links between junction points or made up of micelles linked together by molecular chains inter-connecting the individual micelles. I t i s probable from the results of Alexander and Gray that the latter i s the actual structure. The contri-bution to the viscosity due to hydrodynamic type of resistance to motion of the micelles would be small compared to the viscous effect contributed by rearrangement of chemical bonds so i t i s doubtful i f this difference could be determined from measurements of the type made with the Visco-elastometer alone. 56 In summary, from the experimental data obtained i n this work i t i s found that both r i g i d i t y and viscosity vary with about the third power of the concentration. The distribution of relaxation frequencies i s nearly independent of frequency and the peak i n the relaxation spectrum i s at an angular frequency of the order of 0.003 seconds" 1 or conversely the relaxation time i s of the order of 300 seconds. This relaxation time i s very long and i s detectable only by relaxation or creep methods since dynamic methods would not be practical at such low frequencies. I t i s concluded from the experimental results that the structure i s three dimen-sional and that actual bonds must be broken and reformed during relaxation and steady flow. Future work with the Visco-elastometer should include the determination of temperature dependence and the measurement of the relaxa-tion after sudden cessation of steady flow. Neither of these experiments are possible with the instrument i n i t s present form. To make measurements at different temperatures i t w i l l be necessary to enclose the instrument completely and use an atmosphere saturated with solvent. .A. new transducer c o i l w i l l be necessary for this purpose since neither the insulation on the wire nor the c o i l impregnating material i s resistant to the solvents used. To make measurements of stress relaxation after cessation of steady flow the b a l l bearings carrying the outer cylinder must be replaced with sleeve bearings. With the small dimensions used and the high sensitivity required b a l l type bearings do not appear to be satisfactory. I t i s hoped that the two above mentioned experiments w i l l supply additional information from which further conclusions can be made about the molecular structure of the gels. 57 l e t APPENDIX I A Method of F i t t i n g Two Exponential Terms  to Four Points at Equal Increments To f i t y =/f e -t&G to four points at equal increments, Changing the variable to so that X = O, /j 2t 3 a„c/ K= Xa y X S gives where & - AG*** and $ = /?e*** then (2) These are four equations i n four unknowns, , , -2^ , , , and 2L/ and -2^ are the roots of the quadratic equation 53 Z £ + / > z l + /> = o C5) Multiplying equation (5) by o< ^ and b y , ^ ^ and adding gives Putting /?- o gives These two equations can be solved for ^  and /c? which can then be substituted i n equation (5) to give z, and . Equations (1) and (2) can now be solved for & and and = A e / ^ e This method i s a simplification of Prony's Method which may be found i n Whittaker and Robinson1** 59 BIBLIOGRAPHY (1) Perry, J.D. and Grand!ne, L.D., Jour. App. Phys., 2A> 679 (1953). (2) Nolle, A.W., Jour. Pol. Sd.,, V, 1, (1950). (3) Alfrey, T., Mechanical Behavior of High Polymers. Interscience Publishers Inc., N.Y.., 1948. (4) Rideal, E.K.,: and others, Proc. Roy.. Soc.., A20Q. 135 (1950). (5) Van;Wazer, J.R. and Goldberg, H., Jour. App. Phys., l g , 207 (1947). (6) Goldberg, H. and Sandvik, 0., Anal. Chem., 12, 123 (1947). (7) Gunn, G.B., unpublished results. is) Flynn, J.T., M.A. Sc. Thesis, University of Bri t i s h Columbia (1951). (9) Ferry, J.D., Rev. S d . Inst., 12, 79 (1941). U 0 ) Ter Harr, D., Physica, X7I, 719 (1950). [ l l j Maxwell, J.C., P h i l . Trans.. Roy. Soc., London, 157. 49 (1867). [IZl Sips, R.,, Jour. Pol. Sd., V No. 1, 69 (1950). [13) Zener, C., E l a s t i d t y and Anelastidtv of Metals. ^14) Whittaker and Robinson, Calculus of Observations. [15) Markovitz, H., Jour. App. Phys., 2 J , 1070 (1952). ;i6) Oldroyd, Strawbridge and Toms, Proc. Phys. Soc., B 6A. 41 (1951). .17) Mason, W.P., Electromechanical Transducers and Wave F i l t e r s . D. Van Nostrand (1948). ~ v."' . . ,18) Ferry, J.D., Sawyer, W.M., and Ashworth, J.N., Jour. Pol. Sd., g, 593(1947). 19) Sneddon, Fourier Transforms. 20) Gunn, G.B., Ph. D. Thesis, McGill (1950). 21) Smith, Ferry and Schremp, Jour. App. Phys., 2Q.t 144 (1949). 22) Philippoff, W., Physik. Z., 884 (1934). 60 (23) Markovitz, Yavorsky, Harper, Zapas and De Witt, Rev. S d . Inst., 2J, 430 (1952). (2A) Ferry* J.D., Jour. Am. Chem. So c , 22, 3746 (1950). (25) Rouse and S i t t e l , Jour. App, Phys., 690 (1953)* (26) Shaffer, H., Can. Jour. Res., B- 26. 481 (1948). 

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